= Polymer scattering function
{title2=$g(0)=N$}
= Single-chain scattering function
{synonym}
For $N$ equal scattering sites on one <polymer>, define $g(\mathbf k)=N^{-1}\sum_{m,n}\langle e^{i\mathbf k\cdot(\mathbf R_n-\mathbf R_m)}\rangle$. It has $g(0)=N$; its normalized form factor is $g/N$. Under a rotationally invariant joint distribution, each separation vector has a uniform direction at fixed magnitude $r$. The angular average is $\tfrac12\int_{-1}^{1}e^{ikr\mu}\,d\mu=\sin(kr)/(kr)$, with value one at $r=0$. Thus $g$ depends only on $k=|\mathbf k|$, and $g(k)=N^{-1}\sum_{m,n}\langle\sin(k|\mathbf R_n-\mathbf R_m|)/(k|\mathbf R_n-\mathbf R_m|)\rangle$. The diagonal self terms are essential in a finite chain and give the large-$k$ limit one for a discrete <Gaussian chain>.
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